Calculating device



Aug. 14, 1951 P. M. PEPPER 2,564,227

CALCULATING `DEVICE Filed June 16, 1947- 3 Sheets-Sheet 1 IN VEN TOR.

TTR/VEYS Aug. 14, 1951 P. M. PEPPER 2,564,227

CALCULATING DEVICE Filed June 16, 1947 3 Sheets-Sheet 2 lEE:

s fa 8 5 76 2 F l 15E INVENTOR. 84M M/Df/D/Dff? Aug. 14, 1951 P. M. PEPPER CALCULATING DEVICE 3 Sheets-Sheet 5 Filed June 16, 1947 FTEJE INVENTOR. P404 /K/fP/Df BY www Patented Aug. 14, 1951 UNITED STATES PATENT l QFFICE CALCULATING DEVICE Paul M. Pepper, South Bend, Ind.

Application June 16, 1947, Serial No. 754,934

22 Claims. l

This invention relates to improvements in calculating devices, and more particularly to an all-purpose calculating device of the slide rule type.

The primary object of this invention is to provide a device of this character by means of which it is possible to make computations accurate to ve decimal places.

A further object is to provide a device of this character comprising a disk having an yannular equal division scale and a gang yof helical logarithmic scales each having a plurality of convolutions, preferably ten in number, to which is centr-ally pivoted two transparent arms having radially extending graduated scales intersecting said disk scales, wherein the graduations are adapted to register with the helical scale in one position of the arms.

A further object is to provide a device of this character having a disk and arms pivoted thereto and extending radially thereof with guide means adapted to accurately position one of the arms at a starting position by feel and without requiring observation of the position of the arms.

A further object is to provide a device of this character having a disk provided with helical scales on both faces thereof, said scales being correlated whereby each thereof is a continuation of the other and the two constitute a logarithmic scale, wherein two sets of transparent arms are provided, one juxtaposed to each of the faces of the disk, and one arm of each set being correlated and held in register with a corresponding arm of the other set, and wherein a novel boltable clutch arrangement is provided.

A further object is to provide a device of this character including a disk having a helical scale to which a transparent arm is pivoted to intersect the convolutions of the scale, wherein said arm is provided with a pointer Ior marker and shiftable longitudinally of the arm to indicate any given position thereof, and which marker is so positioned that it clears and does not inten fere with the pivoting of a second correlated pivoted arm.

Other objects of the invention are to provide rapid computation of an angular position by means of the provision of generalized locating indicia upon the hub of the device, the correlation of the zero position of the number scale with one end of the gang of helical logarithmic scales, the provision of Vernier means on at least one of the arms, and the provision of la device which is sturdy, simple in construction and easy to operate and to read. i

Other objects will be apparent from the following specification.

In the drawing:

Fig. 1 is a face view of one embodiment of the invention.

Fig. 2 is an enlarged view illustrating a portion of the helical logarithm scale.

Fig. 3 is an enlarged fragmentary detail view illustrating the number scale.

Fig. 4 is an enlarged view illustrating the manner in which the device is used.

Fig. 5 is a fragmentary sectional view taken on line 5--5 of Fig. 1.

Fig. 6 is a detail view illustrating the hub marking.

Fig. 7 is a fragmentary perspective view of one of the scale arms illustrating the provision of a. marker mounted adjustably thereon.

Fig. 8 is an enlarged fragmentary view similar to Fig. 5, illustrating a modified construction oi the invention adaptable to devices wherein scales are applied to both faces of the disk and two cor.

related sets of arms are associated with the device.

Fig. 9 is a fragmentary face view illustrating the application of the invention to a slide rule of the elongated ruler type.

Fig. 10 is a view illustrating the provision of a Vernier scale upon one of the arms of the device.

Fig. 11 is a fragmentary face view illustrating the construction for reading one condition from a scale setting made for a second and different condition.

Fig. 12 is a fragmentary view illustrating a. modified embodiment of the invention utilizing a helical marginal number scale.

Fig. 13 is a fragmentary enlarged face view illustrating the manner in which Vernier readings are made with a Vernier scale of the character illustrated in Fig. 10.

Referring to the drawings which illustrate the preferred embodiment of the invention, the numeral 20 designates a disk which is preferably formed of rigid sheet material, such as metal, plastic or like material. The disk 20 has a circular outer edge 22 which is interrupted at one limited portion thereof by a tab portion 24 projecting radially outwardly therefrom. This tab portion has substantially parallel side edges which extend parallel to a radius of the disk centered thereat.

The disk 20 has a central aperture 26 through which extends the shank of a bolt 28 provided with a flat head 30 at one end. A disk or gasket 32 of friction material bears against one face of the disk 20, and one end of an elongated arm 38, which is formed of rigid transparent material, such as a plastic material, is provided with an opening through which the shank of the bolt 28 extends snugly to provide a pivot for said arm 38. A second disk 36 of friction material bears against the outer face of the arm 38, having a central opening fitting around the bolt 28, and a second elongated arm 34 formed of a transparent material, such as a plastic, has an opening tting snugly around'the bolt-.shank28iand bearing Y against the`v friction disk 36. A concavo-convex resilient disk 40 ts around the shank of the bolt 28, andV a nut 42 is threaded on the end of the bolt to apply pressure to the spring washer 4U, whereby the parts, namely, the arms 313 and 33 and the disk 26 are drawn together" inA firm face engagement with the friction f disks 32 rand interposed therebetween. The parts are so constructed and arranged, and the pressure applied is so adjusted, that the arms 34 and 38 may be SWllngito any desired radial ,position upon' the disk and adjusted relative togeachotherasdesired but will remain in any positionat which they;are set. by. the` friction `clamping action applied by the bolt, the nut and the spring washer 4,6.' ,The-armd is of-#a length to terminate adjacent the peripheralrfedgefZZ-f ofthe disk, as best illustrated-inFig. 1. Theearm 13,8 is preferably greater in length and iszadaptedto overlie a portion of the; tab'24,y as illustrated in-lig.A 1. It will be observed that'the terminal :portion 440i the armee is ofv the; same vwidthr as the tab 24-so that the user m-ay accurately position; thegarmz38at a predetermined radius:of'ithefdevice -by -simply engaging the opposite edges of the arm extension 4.4aand the tab 24gtogeffectfregistration therebetween.

The disk 2e is;providediwithamarginal annular scaled' which has; a5 .plurality-.01. .eluSpaced graduations. ,inthepreferred.formthe-scale 45 has onehundred` equispa-ced ,primary numbered graduations-43,;the:zero graduation .'50 -of'V which is on the same radi-us,esther-radialicenter of the tab 24. A lplurality of 'secondary figraduations "52 divide equally, and -preferably `into ten Yequal spaces, the portion of theseale:betweenqadjacent primary graduations 148. A jhelical logarithmic scale, .preferably 'having ten f-convolutions, -occupies ,the major portionaofthe-.area of the disk with its outer ends-paced inwardly from the number scale 4.6,-,and itsfinnerendxspaced outwardly of the spring washer 46, so that it is fully visible Athroughout its extent. VThe :logarithmic scaleY isV preferablyqof .the character-bestA shown in Fig. 2 and comprises-.1a ,f helicalrlogarithmic scale 5d-whose outer-end isvradiallyfaligned with the zero-position' ofxthe numberscale. The logarithmic numberscale54 isf-divided by suitably numbered primary scale divisions or: graduations 5.6,; and 4the .spaces between these primarynumbered graduations56aredividedfbythe secondary logarithmic, graduations .1511 intofan equal number ofwidivisionswhich: arezzpreferably ten in number orisorne'multipleor sub-multiple of ten. A second helical scalelinef58gisjuxtaposed to the scale 54and preferablyhasjthe vsame number of con- Yolutions although it Yma-y-ha-vemore :or fewer eonvolutions-t-hanscale 34. :At one side of this sealefmayzbe provi-ded graduations '60 suitably ynumbered which constitute a sine-cosinelogaoflogarithmic'.scales154, :.58 are preferably arranged upon the disk, as illustrated generally at Fig. l. The inner ends of the helical scale preferably terminate outwardly of the spring washer 40 a suiicient distance to permit the application thereto of a third graduated scale 64, best shown in Fig. 6, which is preferably divided into ten equally spaced numbered graduations 66 with respect to which the lines 68 on the arms 34 and 33 may be read at a glance to secure readings of tenths of afull turn of said; armeabout the disk instead -of necessitating `the reading thereof on the large scale 46. y

`--lilach of the arms 34 and 38 is provided with Yan elongated central longitudinal scale line 68 which is of a length and so positioned that it will intersect-each of the scales 46, 54, 68 and 64 upon the disk. Each of the scale lines 68 upon the arms extend radially of the disk, and the line 68 upon the arm 38 is so positioned that when the outer end portion 44 of the arm 38 registers with the2tab =24..,upon.the disk, .thel-ine gffwill intersect the scale/46 at its zero position 50, the scales `54 and 53 4 at ytheirouter ,-end portions, which will be the 1 of the scale 54, zero of the-log cosinev scale"l ofthe log sinesscalefand sof the -log A.tangentecotangent scale, rand the zero position of..thescale64. Each of the scale7 lines 63 is provided with ten numbered? equispaced graduations i6 which-are-so positioned thereon that they register'with the .suceessiveconvolutions ofthe scale 5d when-said arms arellocated at the zero setting, that isat theposition at which thearmf38 isshown in Fig.A l.

By the use ofthe device it is possible to make computations accurate torve signicant'gures of the following types: (a) .'logarithms vof-numbers and :antilogarithms of numbers in thev powers of ten, (b) logarithms of trigonometriciunctiens and antilogarithms of trigonometric functions, and (e) computations of'natural'trigonometric functions. Thusit ispessible-to use the: device asg a mathematical operatorto multiplyor divide anyl two numbers crtheir reciprocals, multplypr divide a number-either real ,or Vcomplex by .fia trigonometric function, and to:multiplyforgdivide any trigonometricvfunction by anotheritrigonometric function. As an illustration :of the manner in which the device `can-,be usedfforsaccuracy to ve points, reference,maybehadto Fig. 4. We will assume here thatthe problem is tofdetermine the logarithm of the'zcosine vof 5,2 degrees. Thearm 38 is setwith tscenterlongitudinal mark coincidingr with the152 degree mark on the helical scale viias read from the cosine graduation 60, as seen in: Fig. 2. It will be seen that thehelical line will then occur. between the divisions l and -8 Vof the scale 1D on` the arm. This will show that ,the logarithm ofthe vcosine of '52 degrees .is '.7 plus. Then byreading from the equal division number. scale .-346 1 it will Ibe observed thatA the'line 468 intersects'the number scale 46 between the divisionsfnumbered 89 and S0, as shown in Fig. 3.Y This gives the second and third places of the reading, namely,.a reading of .789. The scale graduations 52 (Fig. rI3), being divided into'tenths, permits the reading of the next number from the scale, namely, the number 3. The fth placenumbercan be readJ by computing thenearest tenth -portion or fraction between the graduations 52. 'VVhilethe drawing is not exactly accurate,v it is possible with the actual devicerto determine accurately that the mantissa of the logarithm of the cosine 52 degrees is .78934.

'In the use of the device for multiplication purposes the arm 38 is set at zero-and thearm '34 is then set at the proper position with its center line 68 on one of the multiplication factors on the selected one of the helical logarithm number or logarithm trigonometric scales 54 and 58. The two arms so set at selected angularly displaced position are then rotated while that angular displacement is maintained until the first arm 38 is set with its center line on la second multiplication factor on the same or a second selected helical scale 54, 58 and any result is read from any selected helical scale with reference to the second arm, that is, the arm 34. This permits multiplication of two numbers, or of a number by any trigonometric function, or of any two trigonometric functions to make computations in spherical trigonometry and to make possible the solution of problems involving compound angles, as easily as computations involving ordinary numbers. The index of the answer, i. e., the turn of the helical scale 54-58 on which the answer is to be read, is the sum of the indicia of the various factors for multiplication. For example, to multiply three by four, the center line 68 of arm 38 is set on register as shown in Fig. 1, The center line 63 of arm 34 is set on 3 of the logarithm number scale 54, noting that the index of 3 as read from scale 18 where center line 68 of arm 34 intersects said scale is 4.8. The two arms are then rotated as a unit, i. e. maintaining their exact angular displacement relative to each other, until the center line 68 of arm 38 intersects the scale 54 at 4. The index of 4, as read from scale on arm 38 Iat said intersection is 6.0. The sum, 10.8, of the indicia readings of the settings for "3 and 4 is the index of the products desired. If the scale has only ten turns, and the index sum is greater than ten, the index reading is reduced by ten (or a multiple thereof if the index sum of three or more numbers being multiplied exceeds 20), and the product is read from the scale 54 at the intersection of line 68 on .arm 34 with the rst turn of the scale, i. e., at index sum ".8. In the case of multiplication of two trigonometric functions, to solve a formula such as cos C=cos A cos B from spherical trigonometry, the settings are 'made on scale 58, the indicia 10 being read against scale 54, i. e., where the lines 68 intersect the base line of scale 54 contiguous to their intersections with scale 58. Otherwise, the procedure is similar to that for number multiplication. It will be apparent that the logarithm trigonometric scale permits direct reading of angles without need to convert angle readings to numbers and to reconvert numbers to angles.

Where division is to be performed, the arm 38 is rst set on the denominator and the arm 34 is set on the numerator, and the index readings on scales 'I0 are noted; the two arms are rotated in unison, maintaining their relative angular displacement until the line 68 of arm 38 registers with "1 of the logarithm number scale; and the quotient is read at the intersection of line 68 of arm 34 with the turn of the scale 54 indicated by the difference of the index readings, i. e., the index of the denominator subtracted from the index of the numerator as noted in the initial setting. If necessary, the index reading of the numerator is increased by ten to make it exceed the index reading of the denominator before subtraction.

For the purpose of enabling the operator to have an indicator of the radial position of any setting directly upon the instrument, thereby enabling the operator to avoid the necessity for penciled notations in conjunction with the use of the device, I propose to use an indicator of the type best illustrated in Fig. '7. In this construction it will be observed that a dovetail groove 'I2 is formed in the arm which may be either the arm .34 or the arm 38 or both, said dovetail groove being positioned alongside and parallel to the scale graduations 10 upon said arm. An indicator member, preferably comprising a sheet metal stamping, a plastic molded part or any other item which may fit snugly but slidably in the dovetail groove 72, is provided therein. This member, here shown as a sheet metal member having diverging end portions 14 fitting in the dovetail groove and a transverse central portion 16 preferably flush with the face of the arm 3'4, may be provided with a suitable mark which can be set at register with any selected position of the graduated scale 10. The function and advantage of a slide of this character is believed to be apparent and enables the use of the device in any convenient manner.

In cases where it is desired to use a Vernier scale, the same can be applied to one or both of the two arms 34, 38 in the manner illustrated in Fig. l0 where the Vernier marking lines '16 extend alongside the marking line 68 on said arm, being juxtaposed to graduations I0 as shown and intersect the selected helical scale 54 or 58 which will include the necessary Vernier graduations or markings properly correlated with the arm markings 16. Such a Vernier scale is thus juxtaposed to the computing scale on the arm and is used in the same manner as Vernier scales are used conventionally.

It is possible to apply the same larrangement and combination for computation purposes to a conventional slide rule, such as is shown in Fig. 9, having a longitudinally grooved body portion 80 and one or more slides 82 mounted in the groove or grooves of the body 88. In this case the graduated scales 84 upon the body 8D and the slide 82 are preferably inclined slightly with reference to the longitudinal dimension of the scale so that the end of one scale line at one end of the rule is at the same transverse position as the end of the next adjacent scale at the opposite end of the rule.. Each individual scale line 84 corresponds to a selected portion of a full range of the scale. A cross-slide member 86 of conventional character formed of transparent material is provided with a central marking 88 with graduations 9U thereof spaced equally to the spacing of the rule scales 64 Iand adapted to accurately register therewith at one position of the device. It will be apparent that slide rule computations may be made from this device using the principle and in substantially the same manner as hereinabove described with reference to the circular or disk type. Specifically, one manner in which such a rule can be used is to adjust the slider 82 to zero position, then adjust the cross-slide 86 for registration of its hairline 88 with the rst factor of the computation, noting the index as taken from scale at the intersection of line 88 with the row or section 84 at which said factor occurs, then moving the slider 82 to the right or left for end registration thereof with the hairline 88 of the cross-slide 86, then moving the cross-slide 86 for registration of its hairline I88 with the second computation factor, noting from scale 90 the index of the row in which that factor occurs, and then moving the slide 82 back to zero position to read the result on the slide 82 at the hairline 88 of the cross-slide 86 on that row or section 84 aangaan? of the rulefwhoseindexfas taken from: scale-ill)` isthetsumfofthe indices of therowstllzon which theitwogfactorsLoccurred. -Ifthe lines orzsca'les 8d arefnot inclined, and ifthenumberfof lines 'rfequals the number `of fgraduations1-9U, then whenv rightregisteraof` the slide is used; theV product isfoundfat theifsum of thein'diceszofthe factors plus 1. Ihisaoperation maybe simplified in cases-where thefscales'upon the slidesl are duplicated upon ,the bodyi andthe scaleof vthe crossslid-e is: correspondingly* set up tov provide a scale cooperating Withthescale uponzthe body'anda second scalecooperating with theA scale upon'the slide. "In that instancetonlythreeoperations are necessary, na-melyfsetting the cross-slide 86 'at the firstzfactor .onthe body 8D and noting the indexofthe'rstffactor' as read from that scale 9B of the cross-slideifwhich"cooperates Awiththe body til, shifting-thesslide'lZ- to lsuitable register, either right or fleftywith the cross-slide hairline 88, yand shifting Vthe cross-slide 86 to' theisecond factor read from the properline Be'offthesslide 82. The product may then be read fromthe body 80 at the intersection of hairline 8B with that rowll whose'index isequal to the'sumzof the indices of the `rows-of the two factors, or, if right-hand register 'hasbeenused at the row whose index isthe sum `of the indices of the rows of the two factorsplus'l.

Y In someinstances it may beldesired to divide the logarithmicf'scale.into two parts in order to secure the desireddegreev of accuracy. For this purpose one-portion of .the helical scale-513, 58 may-be placed .upon one face of the disk yand the balance thereof placed upon the otherface `of the disk. In such instances thescales will preferably each bearranged toshave-.ten vcomplete convolutions uponeach face. .v In .firder'to'enable readings to be transferred .from one escale face to the other, -a construction of thetype of Fig. 8 is used. In this form thedsk 20 has'a large opening 92 formed .at the centerthereof. Within this openingis positioned an-annular'memberllll yof slightly greater thickness and having asnug but freely sli'dable i'it inthe opening-92. TheannularA member V9d is of slightlygreater thickness than the disk 20 sothat a pair of armsQB-secured to oppositeA faces of the annular members@ as by means of-,pins S8, have clearance with thedisk 20. -A pair of..annular friction members are interposed between the .arms-96 and the disk20 at opposite faces thereof. yWithin theannularmember all is positioned Ta hub member HB2 of greater thickness than the Vcombined thickness of the arms 9B. and the annular member 94, and to this hubby means vof thepins |04 yare secured-two radialarms 466. lirictionfdiskV members 08 are interposed between .the arms 55 :and Ille. A bolt MFL passes through a .central'opening -in the hub N12 and the arms |96 .and Vthrough concavo-oonvex spring vdisks l l2 positioned in engagement with opposite arms IBB. The spring disks H2 are 4urged inwardly by the pressure exerted between `the .heacl .l I4 and a nut l l upon the .bolt `I lll. .In `this connection, assuming the scale` at one face would be a helical scale covering a given .limited range of the-log sine-.cosine and log tangent-.cotangent for angles near zero degrees, vitis possible to use a single set of graduations instead of separate graduations 60-62as shown, by applying two longitudinal lines, instead of one line 68, as shown, which lines are arranged in proper relation to each other .and to helical scaleline .-to designate respectively the logsineor .cosine and .the log tangent orcotangent tat Aany given settingof the 1 arm. One example is shownfin lFig. 11 which illustratesfza construction rwhich enables a vlimited extent 1. of logarithm sine-.cosine scale to be used in making computations which ordinarily require a logarithm tangentecotangent scale. As thereillustrated, if the tangent of 9-27.5 is desired, the ,curvedline 69ris set against'the logarithm sine scale 60 `at the angle desired, it being Yassumed that tthe logarithm tangentecotangent scale t2 is omitted, and thepoint of intersection of' the straight line 68 `with l'the corresponding -convolution of the logarithm number scalei54 designates the value of the natural tangent, i. e., .16661. In performing a vcomputation involving the tangent of 927.5, the setting made fromthe logarithm sine scale aseabove mentioned is'used inconjunction with other-setting, as described `above for both'multiplication and division, with the productor quotient -.being read Yfrom the intersection of the straight'linet with the proper'convolutiomof the logarithm number scale "54. If the-product or quotient is'the tangent or cotangent vof Aan angle, the angle is read at the intersection ofthe curved line-69 lwith-*the logarithm sine scale. -It will be apparentthat this construction permits tlie'two arms v to be adjusted to any selected position at which they are held by the friction member '160, said Aarms always being in register by virtue of 'their interconnection by the pins 43S. Similarly the twoarms It areY always held in register and may be adjusted to any radial position at which theyare heldwby the friction members l'. Consequently, when a settingihas been made of the arms with reference to the scale atlone face Yof thedevice, that reading maybe transferred to'the other scale on the other face of the device bythe opposite arm-whose'radial position .will -exactly coincide `-with the radial position of the rst mentioned device.

-While `the equalrdivision scale flfshas :been describediabove as annular'in'form, it may behelical -and.composed of a plurality of convolutions, as shown ated? in Fig. 112, wherein the divisions 53 Iare equiangularlydisplaced from the center of vthedisk 20. Inthis casela second group :of graduations -l I corresponding 'to graduations r'Hl but correlatedwith the number scale only, will belplaced on the farms Sil-38 at the outer Vend of line68 thereof. rlhus, if'the equal division scale 'lishelical and composed of three convolutions, the centerline or hairline' of the arm, and 684 of the arm V34,: may be provided with three intersecting graduations 7l, of which at least :one will register with -a'convolutionof the `helical scale 41 when the center lines of said arms registerwith the zero position ofY said scale 47. This setof 4graduations will obviously be spaced longitudinally from the setof gra-duations 70 shown and may v.be numbered vvto correspond withthe numberofv convolutions of the scale 45 or in-any other manner.

.While thepreferred embodiment vof the invention hasbeen illustrated and described herein, it will .be'understood that the constructioniof the device may begaltered within the scope of theappended claims without departing from-the spirit of the. invention.

I claim:

1. A calculating device comprising a disk'hav ing an annular equal division scale and afhelical logarithmic scale correlated-at a reference point with said first scale and lhaving la plurality of convolutions spaced'equallyon all'ra'dii, a'pair of transparent arms pivoted at the center oisaid disk, friction means for vholding said arms in selected angular relationl said arms each having a radial mark and a plurality of equi-spaced cross marks adapted to register with said convolutions when said radial mark registers with said reference point, said disk having a radial outward projection correlated with said reference point and of the same width as one of said arms, said arm being of a length to project beyond the periphery of said disk, said projection .being positioned clear of the plane of rotation of said arms whereby said last named arm may overlie said projection.

2. A calculating device comprisinga .disk having a helical scale having a plurality of convolutions spaced on all radii and having graduations spaced differently on different convolutions according to a selected mathematical law, a pair of transparent arms pivoted at the center of said disk, friction means for holding said arms in selected angular relation, said arms each having a radial mark and a plurality of spaced cross marks adapted to register with said convolutions when said radial mark is in predetermined vrelation to said helical scale, one of said arms having longitudinal Vernier markings extending alongside said radial marking and -curvedto vary the spacing from said radial marking according to the spacing of said convolutions and the graduations thereof.

3. A calculating device comprising a unit having a mathematical scale comprising a Iplurality of portions extending alongside each other in equally spaced relation, a shiftable transparent member carried by said unit, said member spanning said unit and having a graduated scale intersec ting said first scale with its graduations reg- 1" istering with the portions of said first scale in one predetermined adjustment of said mem-ber, said scales being so positioned relative to each other and to the path of movement of said member that said graduations are displaced from register with said iirst scale in proportion to the spacing of said member from said predetermined adjustment on said unit, and guide means in the plane of and projecting from-said unit and having an edge portion adapted to register with an edge portion of said member when the latter is set at said predetermined adjustment.

4. A calculating device comprising a unit having a gang of helically arranged sets of scale marks extending alongside each other in spaced, juxtaposed, interlayed relation and at least one helical line interlayed with said sets of scale marks, a shiftable transparent member carried .by said unit, said member spanning said unit and having a graduated scale intersecting said scale gang and line with its graduations registering with said helical line in one predetermined adjustment of said member, said scales being so positioned relative to each other and to the path of movement of said member that said graduations are displaced from register with said helical line in proportion to the spacing of said member from said predetermined adjustment on said unit.

5. A calculating device comprising a rigid disk having a marginal helical graduated scale having a plurality of convolutions and an inner helical graduated scale having a plurality of convolutions and terminating at one end in radial alignment with the zero marking of the first helical scale, a pair of transparent radial arms each having a graduated longitudinal scale juxtaposed to land intersecting the first helical scale and a seclond graduated longitudinal scale juxtaposed to and intersecting said inner helical scale, with 10 the zero graduations of the arm scales registering with said convolutions at zero setting.

6. A calculating device comprising a rigid disk having a circular graduated scale,- a pair of transparent arms pivoted at the center of said disk and having a longitudinal scale intersecting the disk scale, and a. guide portion projecting from said disk substantially in its plane and adapted for registration of an edge portion of an arm with an edge portion thereof for setting one arm in a selected radial position.

'i'. A calculating device comprising a rigid disk having a circular graduated scale, a pair of transparent arms pivoted at the center of said disk and having a longitudinal scale intersecting the disk scale, and a radial projection carried lby and substantially in the plane of said disk, one of said arms being elongated and terminating in an end portion of the same width and adapted to overlle and register with said projection, said projection being positioned to effect a predetermined relation of said scales when said arm registers therewith.

8. A calculating device comprising a rigid disk having a circularly arranged measuring scale on each face thereof, said disk having a central aperture, an annular member rotatable in said aperture, a pair of transparent arms xedly secured to said annular member, said arms being positioned in register at opposite sides of said disk, a central member mounted rotatably in said 'annular member, a second pair of transparent arms iixedly secured to said central member, said last named arms being positioned in register at opposite sides of said disk in planes spaced outwardly of the planes of said first arms, and means for holding said annular member, central member and said arms in operative assembled relation to said disk.

9. A calculating device comprising a rigid disk vhaving a circularly arranged measuring scale on each face thereof, said disk having a central aperture, a rotatable unit including an annular yhub mounted in said disk aperture, and a pair of registering arms secured to said hub and positicned at opposite sides of said disk, a second rotatable unit including a hub mounted within said annular hub and a pair of registering arms secured to said last named hub and positioned in planes outwardly displaced from the planes of the rst arms, friction means interposed between said units and between said ilrst mentioned unit and said disk, and clamping means for pressing said units and disk against said friction members.

l0. A calculating device comprising a disk having a graduated helical scale and at least one transparent arm pivoted to said disk at the center of the disk and having two variably spaced lines extending longitudinally thereof and so related to the helical scale, the spacing of the divisions of said scale and to each other that one thereof may be used with reference to said helical scale to designate one type of condition, and the other thereof may be used with said helical scale to designate another type of condition.

l1. A calculating device comprising a disk having a graduated logarithmic sine helical scale of limited range, and a pair of transparent arms pivoted to said disk at the center of the disk, at least one of said arms having two variably spaced lines extending longitudinally thereof and so related to the lead of the helical scale, to the graduations of Said scale and to each other that having a circular graduated scale, a transparent arm pivoted at the center ofsaid dlsk and having' a mark intersecting saidv disk scale, and a `portion projecting from said disk substantially in its plane and, adapted `for registration of an edge portion of said arm with an edge portion thereof for setting said arm in a selected radial position.

13. A calculating device comprising a rigid disk having. a curved measuringscale on each face thereof, said disk having a central aperture, an annular member rotatable in said aperture, a pair of transparent arms xedly secured to said annular member, said arms being positioned in predetermined relation to each other at opposite sides of said disk, a central4 member mounted rotatably in said annular member, a second pair of transparent arms fixedly secured to said central member, said last named arms being positioned in predetermined relation to each other at opposite sides of said disk in planes spaced outwardly o -f the planesof said first arms, and means-for holding said annular member, central member. and saiduarms in operative assembled. relation to said disk.

14.' A calculating device comprising a disk having a helical scale and graduations spaced according to a selected mathematical law, at least one transparent arm pivoted at the center of said disk, said arm having a radial` mark and having longitudinal Vernier markings extending alongside said radial marking ,and curved to vary the spacing from said radial marking according to-` the spacing of different parts of said helical scale from the center of-,said4 disk and the spacingV of scale graduations at different parts of said scale.

A calculating device comprising a substantially rigid sheet member having aplurality of spaced indicia and having.Y an aperture therein, an annular hub rotatable in said aperture and of a thickness slightly greater than said member, a pair of members carried by opposite faces of said annular hub and overlying a portion of said sheet member surrounding said aperture, annular friction members interposed between said sheet member and each overlying member, an inner hub rotatable in said annular hub and of a thickness slightly greater than the spacing of the outer faces of said overlying members, a second pair ofoverlying members carried by opposite faces of said inner hub and overlying a portion of said first overlying member, annular friction members interposed between overlying members of adjacent pairs, and means for pressing said last named overlying members inwardly, at least one overlying member of each pair having indicia correlated with the indicia on said sheet member.

16. A calculating device comprising a substantially rigid sheet member having mathematical indicia on one face, a transparent member rotatable on said firstmember and having indicia correlated with said first indicia according to a predeterminedmathematical law, and a portion projecting from said first member substantially in its. plane and having a reference edge, an edge portion of said second member being adapted for registration with said reference edgeA for setting saidmembers in selected'frotative relation.v

17. A calculating. deviceI comprising a` substantially rigid sheet member having a scale thereon graduated according to a selected mathematical law, said scale having variably spaced graduations a transparent member having a mark thereon intersecting said scale, said transparent member Ybeing shiftable on said first member in a predetermined path to progressively change the point of said mark at which said mark intersects said scale, said transparent member having Vernier markings extending alongside said first mark and spaced from said first mark according to'that mathematical law which is determined by the positions of the graduations-of said scale and to the shifting of the point of intersection of said first mark with said scale.

18. A calculating device comprising a substantially rigidsheet member having a` scale thereon graduated accordingY to a selected mathematical law, a transparent member having a mark thereon intersecting saidvscale, said transparent member being shiftable on said first member in a predetermined path to progressively change the point on said mark at which said mark intersects said scale, said transparent member having a second marking extending alongside said first mark and spaced from said first mark as determined by the difference between a, second mathematical function and the function graduated according to said first mathematical law.

19. A calculating device comprising a substantially rigid sheet member having indicia on each face thereof, said sheet member having an aperture, an annular member rotatable in said aperture, a pair of transparent arms xedly secured to said annular member, said arms being positioned in predetermined relation to each other at the opposite sides of said sheet member, a central member mounted rotatably on said annular member, a second pair of transparent arms xedly secured to said central member, said lastA named arms being mounted in predetermined relation to each other at opposite sides of said sheet member and spaced outwardly of the, positions of said first arms, and means for holding said annular member, central member andsaid arms in operative assembledrelationto said sheet member.

20. A calculating. device comprising a substantially rigid sheet member having indicia on each face thereof, the. indicia on opposite faces being correlated to each other by a selected mathematical law,.said sheet/member'havng anaperture, an annular member` rotatable in said aperture, a pair of transparenty members xedly secured to said annular member and overlying a portion of said sheet-memben said transparent members being positioned in predetermined rela-- tion to eachother at` opposite sides of said sheet mem-ber, acentra-l member mountedrotatably in said annular member, asecond pair'of transparent members fixedly secured-to said centralmember and overlyingfatleast' aportion of said first transparent member, said-'lastnamed transparent members; being positioned inv predetermined relation to Y each other at: opposite sides of said sheet member and spaced outwardly of. the positions of said first transparent members, and means for holding said annular member, central member and said .transparent members in operative assembled relation to said sheet member.

21. A non-logarithmic calculating device comprising a substantially rigid sheet member having a-scale thereon graduated according to a selected mathematical law, a` transparent member hav-ingVY amark thereon intersecting said scale,

' 13 said transparent member being shiftable on said rst member in a predetermined path to progressively change the point of said mark at which said mark intersects said scale, said transparent member having Vernier markings extending alongside said first mark and spaced from said first mark according to that mathematical law which is determined by the positions of the graduations of said scale and to the shifting" of the point of intersection of said rst mark with said scale.

22. A calculating device comprising a substantially rigid sheet member, a linear mark on said member, said linear mark being laid out'according to a selected non-logarithmic mathematical law, a scale along said linear mark, said scale graduated according to a second selectedmathematical law, a transparent member having a measuring mark thereon intersecting said linear mark. said transparent member being shiftable on said rst member in a predetermined path to progressively change the point of said measuring mark which intersects said linear mark, said transparent member having Vernier markings extending alongside said measuring mark and spaced from said measuring mark according to 14 that mathematical law which is determined by the positions of the graduations of said scale and to the shifting of the point of intersection of said measuring mark with said linear mark.

PAUL M. PEPPER.

REFERENCES CITED The following references are of record in the file of this patent:

UNITED STATES PATENTS Number Name Date 486,070 Andrews Nov. 15, 1892 1,207,439 Picolet Dec. 5, 1916 1,211,581 Henoch Jan. 9, 1917 1,404,019 Gilson Jan. 17, 1922 1,435,512 Boggio Nov. 14, 1922 1,436,282 Nuckolls Nov. 21, 1922 2,426,862 MacDonald Aug. 26, 1947 FOREIGN PATENTS Number Country Date 863 Great Britain Mar. 1, 1881 28,603 Great Britain Dec. 11, 1912 180,695 Great Britain Aug. 27, 1923 533,945 France Dec. 23, 1921 

